Topological spin-hedgehog crystals of a chiral magnet as engineered with magnetic anisotropy

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Topological spin-hedgehog crystals of a chiral magnet as engineered with magnetic anisotropy

N. Kanazawa, J. S. White, H. M. Rønnow, C. D. Dewhurst, D. Morikawa, K. Shibata, T. Arima, F. Kagawa, A. Tsukazaki, Y. Kozuka, M. Ichikawa, M. Kawasaki, and Y. Tokura

Phys. Rev. B 96, 220414(R) – Published 28 December 2017

Abstract

We report the engineering of spin-hedgehog crystals in thin films of the chiral magnet MnGe by tailoring the magnetic anisotropy. As evidenced by neutron scattering on films with different thicknesses and by varying a magnetic field, we can realize continuously deformable spin-hedgehog crystals, each of which is described as a superposition state of a different set of three spin spirals (a triple- $q$ state). The directions of the three propagation vectors $q$ vary systematically, gathering from the three orthogonal $〈 100 〉$ directions towards the film normal as the strength of the uniaxial magnetic anisotropy and/or the magnetic field applied along the film normal increase. The formation of triple- $q$ states coincides with the onset of topological Hall signals, that are ascribed to skew scattering by an emergent magnetic field originating in the nontrivial topology of spin hedgehogs. These findings highlight how nanoengineering of chiral magnets makes possible the rational design of unique topological spin textures.

Received 26 October 2017
Corrected 13 February 2019

DOI:https://doi.org/10.1103/PhysRevB.96.220414

Physics Subject Headings (PhySH)

Spin texture

Condensed Matter, Materials & Applied Physics

Corrections

13 February 2019

Erratum

Publisher's Note: Topological spin-hedgehog crystals of a chiral magnet as engineered with magnetic anisotropy [Phys. Rev. B 96, 220414(R) (2017)]

N. Kanazawa, J. S. White, H. M. Rønnow, C. D. Dewhurst, D. Morikawa, K. Shibata, T. Arima, F. Kagawa, A. Tsukazaki, Y. Kozuka, M. Ichikawa, M. Kawasaki, and Y. Tokura

Phys. Rev. B 99, 089902 (2019)

Authors & Affiliations

N. Kanazawa¹, J. S. White², H. M. Rønnow³, C. D. Dewhurst⁴, D. Morikawa⁵, K. Shibata⁵, T. Arima^5,6, F. Kagawa⁵, A. Tsukazaki⁷, Y. Kozuka¹, M. Ichikawa¹, M. Kawasaki^1,5, and Y. Tokura^1,5

¹Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
²Laboratory for Neutron Scattering and Imaging (LNS), Paul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland
³Laboratory for Quantum Magnetism (LQM), Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
⁴Institut Laue-Langevin, 6 rue Jules Horowitz, F-38042 Grenoble, France
⁵RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
⁶Department of Advanced Materials Science, University of Tokyo, Kashiwa 277-8561, Japan
⁷Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

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Vol. 96, Iss. 22 — 1 December 2017

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Images

Figure 1
(a) Schematic illustration of the setup for small-angle neutron scattering. At the rocking angle $ω = 0^{\circ}$ , the incident neutron beam is parallel to the $Si [\bar{1} \bar{1} 2]$ axis. The magnetic field direction is always parallel to the Si[111] axis (the film normal) for rocking scans around the $Si [1 \bar{1} 0]$ axis. (b) Dark-field transmission electron microscopy images (left and middle panels) reconstructed using the reflections $a$ or $b$ (top right panel). The complementary images represent the coexistence of two types of crystalline domains with different crystalline orientations and chirality, and which are related by mirror operation with respect to the (111) plane (bottom right panel). Here, we call them domains A and B, respectively. (c) Schematic illustration of the cubic spin crystal (main panel) formed from a superposition of three orthogonal helices (inset), with their propagation vectors $q_{i}$ ( $i = 1$ , 2, 3) parallel to the respective $〈 100 〉$ crystalline axes. (d) Expected neutron scattering intensity map on the surface of a sphere in reciprocal space with constant radius $q$ , and under the assumption that the cubic spin crystal state [(c)] is realized in the MnGe thin film with an equal population of the two crystalline domains [(b)]. Red dots represent magnetic Bragg reflections, while no scattered intensity is expected in the blue part of the sphere. White and black dots represent the directions of the $〈 100 〉$ crystalline axes of domains A and B, respectively. If the three white dots represent [100], [010], and [001] directions [white arrows in the inset of the left panel of (b)], the black dots correspond to $[\bar{1} 00], [0 \bar{1} 0]$ , and $[00 \bar{1}]$ directions [black arrows in the inset of middle panel of (b)] due to the mirror operation.
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Figure 2
(a)–(d) Data for the 160-nm-thick film: Scattering intensity maps on reciprocal space spheres at high ( $T = 100$ K) and low ( $T = 2$ K) temperatures [(a) and (b)]; $T$ dependence of the magnetization ( $M$ ) at $μ_{0} H = 0.1$ T [(c)]; and the magnitude of magnetic propagation vector $q$ and magnetic period $λ$ [(d)]. (e)–(h) Data set for the 735-nm-thick film. (i)–(l) Data set for the 1800-nm-thick film. Vertical dashed lines on the peaks of $M - T$ curves in (c), (d), (g), (h), (k), and (l) indicate the respective helimagnetic ordering temperatures $T_{N}$ . Green shaded regions in (c), (d), (g), (h), (k) and (l) indicate the temperature ranges with the formation of the triple- $q$ state.
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Figure 3
(a) View along Si[111] of the reciprocal space sphere showing the peak positions of the scattering intensity patterns at $T = 2$ K and $μ_{0} H = 0$ T: blue, green, and red spots denote the peaks observed from the 160-, 735-, and 1800-nm-thick films, respectively. White and black dots represent the directions of the $〈 100 〉$ crystalline axes of the domains A and B, respectively. White and black dashed lines are guides for the eye, indicating the trajectories that scattering peak positions follow with changing the uniaxial magnetic anisotropy in domains A and B, respectively. (b) Thickness dependence of the angle $θ$ between the film normal (MnGe[111] axis) and the $q$ vectors. The thick gray line is a guide for the eye. (c)–(e) Schematic illustrations of triple- $q$ spin states with $θ = 10^{\circ}, 30^{\circ}$ , and $arccos (1 / \sqrt{3}) \approx 54 . 7^{\circ}$ as represented in terms of the hexagonal magnetic cell. Reddish and bluish arrows indicate spins with positive and negative $z$ components, respectively. (f)–(h) Distributions of hedgehog singularities and the emergent magnetic field in the corresponding green box regions of (c)–(e). Regions with large magnitude of the emergent magnetic field ( $| b | > 0.25$ , 1.45, and 4 for triple- $q$ spin states with $θ = 0^{\circ}, 30^{\circ}$ , and $54 . 7^{\circ}$ , respectively) are colored according to the sign of $b_{z}$ : red and blue for positive and negative $b_{z}$ , respectively. Relative size scale among the magnetic unit cells is not presented precisely for the purpose of visibility [(c)–(h)]. Hedgehog [emergent monopole, (j)] and antihedgehog [emergent antimonopole, (k)] spin singularities appear where the spin directions flip, e.g., at the midpoints between the red and blue clusters consisting of up and down moments as exemplified in (i) for the selected region of dashed squares of (e) and (h). Black arrows in (i) indicate the directions of the large-magnitude emergent magnetic fields.
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Figure 4
(a)–(c) Scattering intensity maps on the reciprocal space sphere at $T = 2$ K and $μ_{0} H ∥ MnGe [111] = 0$ , 2, and 4 T. (d) and (e) Hall resistivity at various $T$ . In (e) the data are shifted in the vertical direction for visibility. The yellow shaded regions in (e) represent estimated nontrivial Hall signals, which are attributed to a topological Hall resistivity $ρ_{y x}^{T}$ due to skew scatterings by the emergent magnetic field. (f) Contour mapping of the nontrivial topological Hall resistivity in the $T - H ∥ MnGe [111]$ plane. The critical magnetic fields for the ferromagnetic transition $H_{c}$ [black dots in (f)] are defined as inflection points in the $ρ_{y x} - H$ curves. In the low $T$ range, the transition between triple- $q$ and conical states occurs between $μ_{0} H = 2$ and 4 T, as indicated with green thick lines. The green curve is a guide to the eye for the region of the phase diagram that hosts the triple- $q$ state.
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